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Smooth economic analysis for general spaces of commodities

Author

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  • Accinelli, Elvio
  • Covarrubias, Enrique

Abstract

This paper provides an extended framework to study general equilibrium theory with commodity spaces possibly of infinite dimensions. Our approach overcomes some difficulties found in the literature since it allows the study of the equilibrium when consumption sets may have an empty interior. It also overcomes the need for separable utilities or utilities that satisfy quadratic concavity. The results are based on restricting the mathematical notions of open neighborhoods, continuity, and derivatives at a point, to only those directions that lie within the positive cone. We prove in this setting ``directional'' equivalents of the Sard-Smale and Preimage Theorems. With these tools, we define the social equilibrium set and show that it is a directional Banach manifold. Together with a suitable definition of projection map, this framework allows a natural equivalent to infinite dimensions of the ``catastrophic'' approach to general economic equilibrium.

Suggested Citation

  • Accinelli, Elvio & Covarrubias, Enrique, 2014. "Smooth economic analysis for general spaces of commodities," MPRA Paper 53222, University Library of Munich, Germany.
  • Handle: RePEc:pra:mprapa:53222
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    File URL: https://mpra.ub.uni-muenchen.de/53222/1/MPRA_paper_53222.pdf
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    References listed on IDEAS

    as
    1. Chris Shannon & William R. Zame, 2002. "Quadratic Concavity and Determinacy of Equilibrium," Econometrica, Econometric Society, vol. 70(2), pages 631-662, March.
    2. Covarrubias Enrique, 2010. "Regular Infinite Economies," The B.E. Journal of Theoretical Economics, De Gruyter, vol. 10(1), pages 1-21, July.
    3. Kehoe, Timothy J & Levine, David K, 1985. "Comparative Statics and Perfect Foresight in Infinite Horizon Economies," Econometrica, Econometric Society, vol. 53(2), pages 433-453, March.
    4. Balasko, Yves, 1997. "Equilibrium analysis of the infinite horizon model with smooth discounted utility functions," Journal of Economic Dynamics and Control, Elsevier, vol. 21(4-5), pages 783-829, May.
    5. Elvio ACCINELLI & Puchet MARTIN, "undated". "A Classification Of Infinity Dimensional Walrasian Economies," EcoMod2005 280900000, EcoMod.
    6. Timothy J. Kehoe & David K. Levine & Andreu Mas-Colell & William Zame, 1989. "Determinacy of Equilibrium in Large Square Economies," Levine's Working Paper Archive 46, David K. Levine.
    7. Accinelli, Elvio, 2013. "The equilibrium set of infinite dimensional Walrasian economies and the natural projection," Journal of Mathematical Economics, Elsevier, vol. 49(6), pages 435-440.
    8. Covarrubias, Enrique, 2011. "The equilibrium set of economies with a continuous consumption space," Journal of Mathematical Economics, Elsevier, vol. 47(2), pages 137-142, March.
    9. Chichilnisky, Graciela & Zhou, Yuqing, 1998. "Smooth infinite economies," Journal of Mathematical Economics, Elsevier, vol. 29(1), pages 27-42, January.
    10. Elvio Accinelli, 2003. "About manifolds and determinacy in general equilibrium theory," Estudios de Economia, University of Chile, Department of Economics, vol. 30(2 Year 20), pages 169-177, December.
    11. Balasko, Yves, 1997. "Pareto optima, welfare weights, and smooth equilibrium analysis," Journal of Economic Dynamics and Control, Elsevier, vol. 21(2-3), pages 473-503.
    12. Bewley, Truman F., 1972. "Existence of equilibria in economies with infinitely many commodities," Journal of Economic Theory, Elsevier, vol. 4(3), pages 514-540, June.
    13. Balasko, Yves, 1997. "The natural projection approach to the infinite-horizon model," Journal of Mathematical Economics, Elsevier, vol. 27(3), pages 251-263, April.
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    More about this item

    Keywords

    determinacy; equilibrium manifold: positive cone;

    JEL classification:

    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • D51 - Microeconomics - - General Equilibrium and Disequilibrium - - - Exchange and Production Economies
    • D53 - Microeconomics - - General Equilibrium and Disequilibrium - - - Financial Markets
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates

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